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Properties

Label 16.0.10794315587...7328.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 11^{8}\cdot 17^{15}$
Root discriminant $317.74$
Ramified primes $2, 11, 17$
Class number $260872208$ (GRH)
Class group $[2, 2, 65218052]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9853649041808, 0, 9615126068768, 0, 1891077591304, 0, 103688499024, 0, 2528793520, 0, 32039832, 0, 218042, 0, 748, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 748*x^14 + 218042*x^12 + 32039832*x^10 + 2528793520*x^8 + 103688499024*x^6 + 1891077591304*x^4 + 9615126068768*x^2 + 9853649041808)
 
gp: K = bnfinit(x^16 + 748*x^14 + 218042*x^12 + 32039832*x^10 + 2528793520*x^8 + 103688499024*x^6 + 1891077591304*x^4 + 9615126068768*x^2 + 9853649041808, 1)
 

Normalized defining polynomial

\( x^{16} + 748 x^{14} + 218042 x^{12} + 32039832 x^{10} + 2528793520 x^{8} + 103688499024 x^{6} + 1891077591304 x^{4} + 9615126068768 x^{2} + 9853649041808 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10794315587750478004949812183658374627328=2^{44}\cdot 11^{8}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $317.74$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2992=2^{4}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2992}(1,·)$, $\chi_{2992}(131,·)$, $\chi_{2992}(571,·)$, $\chi_{2992}(1099,·)$, $\chi_{2992}(1937,·)$, $\chi_{2992}(1363,·)$, $\chi_{2992}(2201,·)$, $\chi_{2992}(2905,·)$, $\chi_{2992}(353,·)$, $\chi_{2992}(1187,·)$, $\chi_{2992}(2025,·)$, $\chi_{2992}(1451,·)$, $\chi_{2992}(1585,·)$, $\chi_{2992}(2419,·)$, $\chi_{2992}(2729,·)$, $\chi_{2992}(1979,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{11} a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{242} a^{4}$, $\frac{1}{242} a^{5}$, $\frac{1}{2662} a^{6}$, $\frac{1}{2662} a^{7}$, $\frac{1}{58564} a^{8}$, $\frac{1}{58564} a^{9}$, $\frac{1}{644204} a^{10}$, $\frac{1}{8374652} a^{11} + \frac{3}{380666} a^{9} + \frac{5}{34606} a^{7} + \frac{2}{1573} a^{5} - \frac{1}{143} a^{3} - \frac{6}{13} a$, $\frac{1}{736969376} a^{12} + \frac{1}{2093663} a^{10} - \frac{1}{380666} a^{8} + \frac{1}{69212} a^{6} + \frac{3}{3146} a^{4} - \frac{4}{143} a^{2} + \frac{1}{4}$, $\frac{1}{736969376} a^{13} - \frac{1}{5324} a^{7} + \frac{5}{52} a$, $\frac{1}{21537889538504900704} a^{14} + \frac{51439023}{177999087095081824} a^{12} - \frac{357213507}{505679224701937} a^{10} - \frac{6832590249}{2022716898807748} a^{8} - \frac{12026022487}{183883354437068} a^{6} + \frac{13175874185}{8358334292594} a^{4} - \frac{1874604413}{138154285828} a^{2} - \frac{1021958411}{10627252756}$, $\frac{1}{21537889538504900704} a^{15} + \frac{51439023}{177999087095081824} a^{13} + \frac{111742413}{11124942943442614} a^{11} - \frac{3700757454}{505679224701937} a^{9} + \frac{9228483025}{183883354437068} a^{7} + \frac{7862247807}{8358334292594} a^{5} + \frac{53770120749}{1519697144108} a^{3} + \frac{18596298925}{138154285828} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{65218052}$, which has order $260872208$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 308865.41107064753 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.8.1680747204608.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16$ $16$ $16$ R ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.22.29$x^{8} + 4 x^{6} + 24 x^{5} + 8 x^{2} + 48 x + 12$$4$$2$$22$$C_8$$[3, 4]^{2}$
2.8.22.29$x^{8} + 4 x^{6} + 24 x^{5} + 8 x^{2} + 48 x + 12$$4$$2$$22$$C_8$$[3, 4]^{2}$
11Data not computed
17Data not computed